How to Find the Point of Intersection?

Finding points of Intersection is an elementary exercise in maths. Suppose we have two Functions F (x) and G (x), then how to find the Point of intersection? For this we Set them equals to each other and then find all values of 'x' that make equality correct. Approach we follow to solve such an equation depends on type of algebraic expressions in functions F (x) and G (x). We use methods of Factorization to solve equations with Polynomials and fractional expressions. Similarly, properties of exponential and logarithmic for exponential and logarithmic expressions and Trigonometric Identities for trigonometric expressions can be used.
Let us consider an example to understand it better. Suppose we have two functions given as: F (x) = 3x - 7 and G (x) = x² + 4x – 27. First step is to equate both functions to get the equation: 3x - 7 = x² + 4x - 27.
As we can see that LHS is in simplified form, no more calculations are required here. Let us combine LHS with RHS to get a resultant Quadratic Equation: x² + x - 20 = 0. This is in standard form of a quadratic equation.
We can now factor this equation as: x² + x - 20 = (x - 4) (x + 5), and then equating factors to zero we get: x = 4 and x = -5.
Substitute values of ”x” in original function equations of F (x) and G (x) to verify that they have same corresponding 'y; values. In this example, at x = 4 and x = -5 we get y = -5 and y = 22 from both the equations. Thus their point of intersection is: (-5, -22).